


Algebra 1

by ChoirGirl



Category: Original Work
Language: English
Status: In-Progress
Published: 2021-03-17
Updated: 2021-03-23
Packaged: 2021-03-25 14:35:08
Rating: General Audiences
Warnings: No Archive Warnings Apply
Chapters: 9
Words: 3,533
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/30090597
Author URL: https://archiveofourown.org/users/ChoirGirl/pseuds/ChoirGirl
Kudos: 1





	1. Expressions, Equations, and Functions

### 1-1 Variables and Expressions

  * Algebraic expression 
    * Consists of sums and/or products of #'s and variables
    * Do not have an = sign
    * Only evaluate when directions say to
  * Variables 
    * Any letter used to represent an unspecified number or value
    * Typically lowercase letters
  * Term 
    * A variable, a #, or the product/quotient of the two
  * Factor 
    * Quantities being multiplied
  * Product 
    * Answer to a multiplication problem
  * Power/exponent 
    * Indicates the number of times the base is used as a factor



### 1-3 Properties of Numbers

#### Properties of Equality

  * Reflexive Property 
    * Any quantity is equal to itself
    * Order can't change
  * Symmetric Property 
    * If 1 quantity equals a second quantity, then the second must also equal the first quantity
  * Transitive Property 
    * If 1 quantity equals a second quantity, and the second equals a third quantity, then the first and the third quantities are also equal
  * Substitution Property 
    * A quantity may be substituted for its equal in any expression
    * If a = b then a may be replaced by b in any expression



#### Addition Properties

  * Additive Identity (0) 
    * For any number (a), the sum of a and zero is a
  * Additive Inverse 
    * A number and its opposite are additive inverses of each other and their sum will always be zero.



#### Multiplication Properties

  * Multiplicative Identity (1) 
    * For any number (a), the product of a and 1 is a
  * Multiplicative Property of Zero 
    * For any number (a), the product of a and zero is zero.
  * Multiplicative Inverse 
    * For every number a/b, where a and b does not equal zero, there is exactly one number such that the product of a/b x b/a is 1



#### Other Properties

  * Commutative Property 
    * The ORDER in which you add or multiply numbers does not change the sum or product.
  * Associative Property 
    * The way you GROUP () 3 or more numbers when adding or multiplying does not change the sum or prdouct
    * The orders stays the SAME, the parentheses moves



### 1-4 Distributive Property

  * Distributive Property 
    * Sharing what is outside of the parentheses with each term on the inside of the parentheses.
  * Like Terms 
    * Equal terms that have the same variable raised to the same power
  * *List your answer in order from greatest variable and power to the least variable and power
  * Factor 
    * reverse distributive property



### 1-5 Equations

  * Equation 
    * An expression that contains an equal sign.
  * Solution 
    * An answer to an equation.
  * Replacement Set 
    * A set of numbers which replace the variable in an equation
  * Solution Set 
    * The number's from the replacement set that makes an equation true
  * Solutions of Equations 
    * One solution
    * No solution
    * All real number/infinite solutions



### 1-6 Relations

  * Coordinate system 
    * Used to locate points
    * Formed by 2 intersecting number lines (x and y axis); they intersect at the origin (0,0).
  * To plot a point... 
    * x-coordinate 
      * Start at the origin and move left (-) or right (+)
    * y-coordinate 
      * Start at the x-coordinate and move up (+) or down (-)
  * *Give coordinate with ordered pair
  * Relations 
    * A set or group of ordered pairs
  * Domain 
    * All x coordinates
  * Range 
    * All y coordinates
  * To be a function all x values must be different
  * 4 ways to represent a relation: 
    * Ordered Pairs
    * Table
    * Graph
    * Mapping
  * Independent Variable 
    * Domain values/input
    * Stand alone; doesn't need anything else in the problem.
  * Dependent Variable 
    * Range values/output
    * Depends on the independent variable (IV); can't stand alone



### 1-7 Functions

  * Function 
    * A relationship between input (x) and output (y) values where each x is paired with exactly 1 term.
  * x cannot be repeated
  * Vertical Line Test  

    * If more than one point of the graph touches a vertical line at the same time you do NOT have afunction




	2. Linear Equations

### 2-1 Writing Equations

  * Translate Sentences into Equations 
    * Fifteen times a number subtracted from eighty is twenty-five 
      * 80-15x=25
  * Translate Equations into Sentences 
    * 6z-15=45 
      * Six times z minus fifteen is fourty-five



### 2-2 Solving One-Step Equations

  * Solve an equation 
    * To find the value of the variable that makes the equation true
    * Use inverse operations to isolate the variable



### 2-3 Solving Multi-Step Equations

  * Rules: 
    * Move all variables to one side of the equation; try to keep variable positive
    * Undo addition or subtraction
    * Undo multiplication or division
  * If you have a denominator you will multiply first



### 2-4 Solving Equations with the Variable on Both Sides

  * You will always move the variable with the smallest coefficient



### 2-5 Solving Equations Involving Absolute Value

  * Absolute Value 
    * The distance a number is from zero on a line
    * Distance can not be negative
  * Expressions with Absolute Value 
    * Steps: 
      * Replace the variable with ()
      * Substitute the value of the variable
      * Simplify the absolute value bars
      * Pull number out of absolute value bars
      * Simplify
  * Evaluate Absolute Value Expressions 
    * Rules 
      * If your absolute value equation is set equal to a negative integer the answer is no solution.
      * If absolute value equation is set equal to a positive number you have to find the positive and negative value of your variable.
      * Will have two answers



### 2-6 Ratios and Proportions

  * Ratio 
    * Is a comparison between two numbers by division.
  * Proportion 
    * Two ratios that are equivalent.
  * Cross Product 
    * Multiply diagonally to see if 2 ratios are proportional.
  * When setting up a proportion whatever unit you put on the top of the first ratio **must** be the unit on the top of the second ratio.



### 2-7 Percent of Change

  * % of change 
    * Is the ratio of the change in an amount to the original amount expressed as a %.
  * % of increase 
    * New amount is more than old amount
  * % of decrease 
    * New amount is less than old amount
    * You will have a negative %
  * % of change 
    * (new-old)/old x 100
  * Sales Tax 
    * Change tax to decimal (/100)
    * Multiply decimal by subtotal
    * Add tax amount to subtotal to get grand total
  * Discounts 
    * Change discount to a decimal (/100)
    * Multiply by decimal
    * Subtract discount cost from subtotal to get grand total



### 2-8 Literal Equations and Dimensional Analysis

  * Solve for a Specific Variable




	3. Linear Functions

### 3-1 Graphing Linear Equations

  * Linear Equation  

    * An equation that forms a straight line when graphed
    * x and y have to separate terms
    * x and y cannot be raised to a power greater than 1.
  * Standard Form 
    * Ax+By=C
    * The leading coefficient (A) cannot be negative. If it's negative, divide each term by -1.
    * The leading coefficient cannot be a fraction. If it is, multiply each term by the denominator.
  * X-intercept 
    * Point where a graph crosses the x-axis
    * It is the value of x when y is zero.
    * (x,0)
  * Y-intercept 
    * Point where a graph crosses the y-axis.
    * It is the value of y when x is zero.
    * (0,y)
  * Graph by Using Intercepts 
    * Equation needs to be in standard form
    * Solve for x-int by plugging 0 in for y
    * Solve for y-int by plugging 0 in for x
    * Plot both intercepts and graph your line
  * Graph Using a Table - Graphing by Making a Table 
    * Solve the equation for y. (y=mx+b slope-int form)
    * Plug in -1, 0, 1 for x.
    * Solve for y
    * Plot points and graph



### 3-2 Solving Linear Equations by Graphing

  * Linear Function 
    * A function that graphs a straight line
  * Parent Function 
    * f(x)=x
    * Type of Graph 
      * Line
    * f(x) = y = 0
  * Root 
    * Solution
    * Is the x-int
    * Where the line crosses the x-axis



### 3-3 Rate of Change and Slope

  * Rate of Change 
    * A ratio that describes, on average, how much one quantity changes with respect to a change in another quantity
  * Slope of a Nonvertical Line 
    * The ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run) as you move from one point to another
  * Rate of Change 
    * A ratio (fraction) that describes how much one quantity changes with respect to a change in another quantity
    * Is the same thing as slope (m)
    * ROC = delta y / delta x 
      * Delta - change in
    * Time is always represented by x-coordinate
  * To be a linear function you must have a **constant rate of change** between your x-coordinates and y-coordinates
    * CROS 
      * Goes up or down by the same value
  * Slope (m) 
    * Is a ratio (fraction) that shows the relationship between the rise and the run of a linear graph
    * m = rise / run
    * m = delta y / delta x
    * m = y2-y1/x2-x1
  * Positive Slope 
    * Climbing a mountain
    * As x increases, y increases
  * Negative Slope 
    * Sliding down
    * As x increases, y decreases
  * Slope of Zero 
    * No x-intercept
    * Horizontal line
    * 0/x
    * Slope = 0
  * Undefined Slope 
    * y/0 = zero as denominator
    * Vertical line
    * x = ?



### 3-4 Direct Variation

  * Direct Variation 
    * An equation that illustrates a constant rate of change
    * y=kx 
      * k = constant of variation = ROC = slope
    * k cannot equal 0
    * Graph of y=kx always passes through the origin. (0,0)
    * k > 0 slope will be positive / y=kx
    * k < 0 slope will be negative \ y=-kx
  * Write Direct Variation Equation from Graph 
    * Name coordinates
    * Find slope
    * Write the equation as direct variation y=kx
  * Graph a Direct Variation 
    * Plot a point at the origin (0,0)
    * From there use slope to get your second point
    * Draw your line



### 3-5 Arithmetic Sequences

  * Sequence  

    * A set of #'s in a specific order
  * Arithmetic Sequences 
    * A sequence where the difference between consecutive terms in constant
    * Has a common difference
  * Common Difference 
    * Constant difference between consecutive terms
    * d (common difference)
  * Find the nth term  

    * an=a1+(n-1)d 
      * n = the term you are solving for
      * 1 = first term in the sequence
      * d = common difference



### 3-6 Proportional and Nonproportional Relationships

  * Proportional Relationships 
    * The graph must past through the origin (0,0)
    * Equation can be written using y=kx
    * Constant ROC between x & y coordinates




	4. Linear Functions and Relations

### 4-1 Graphing Equations in Slope-Intercept Form

  * Slope-intercept 
    * y=mx+b
  * List 
    * Graph y-intercept
    * Use slope form y-int get a second point
    * Draw a line that goes through both points
    * Use calculator to check yourself



### 4-2 Writing Equations in Slope-Intercept Form

  * Write Equation Given 1 Point & Slope 
    * Steps 
      * Plug-in what you have into slope-intercept for m and solve for b
      * Rewrite the equation using the slope and y-intercept
  * Write Equation Given 2 Points 
    * Steps 
      * Find slope m=y2-y1/x2-x1
      * Use slope and 1 point to find b.
      * Rewrite equation using your slope (m) & y-intercept (b).



### 4-3 Writing Equations in Point-Slope Form

  * Point-Slope Form
  * Write in Point-Slope Form
  * Slope-Int Form
  * Standard Form



### 4-4 Parallel & Perpendicular Lines

  * Parallel Lines 
    * 2 lines that never intersect and stay the same distance
    * Have the same slope
  * Perpendicular Lines 
    * 2 lines that intersect and form 4 right angles at the point of intersections
    * Slopes are opposite reciprocals; change sign and flip



### 4-5 Scatter Plots and Lines of Fit

  * Scatter plots  

    * Type of graph with 2 sets of data (x and y) plotted as ordered pairs
  * Line of Fit 
    * Line that describes the general trend of plots on a scatter
    * You draw the line of fit



### 4-6 Linear Regression and Best-Fit Lines

  * Best-fit line 
    * A more precise line of it constructed by an algorithm
  * Linear Regression 
    * The name of the algorithm used to find the best-fit line
  * Correlation Coefficent 
    * Is a number that describes the power of the relationship between data plots and states if the correlation is positive or negative
    * The closer to +1 the stronger the positive correlation
    * The closer to -1 the stronger the negative correlation
  * Calculator Instructions 
    * y=ax+b
    * a = slope
    * b = y-intercept
    * r = correlation coefficent
    * Steps: 
      * Input lists 
        * Stat -> edit -> enter 
          * L1 = x coordinates
          * L2 = y coordinates
      * Stat -> over to CALC -> down to Lin Reg (ax+b) -> enter



### 4-7 Special Function

  * Greatest Integer Function 
    * Equation: f(x) = [[x]]
    * Graph: step graph
  * Absolute Value Equation 
    * Equation: f(x) = |x|
    * Graph: V-shaped
  * Piece-wise Defined Function 
    * Equation: f(x) = {-2x if x > 1 ; x+3 if x ≤ 1




	5. Linear Inequalities

### 5-1 Solving Inequalities by Addition and Subtraction

  * Steps  

    * Combine like terms on one side
    * Move variables to the left if possible
    * Undo addition or subtraction
    * Undo multiplication or division (if you multiply or divided by a negative number, FLIP your sign)
    * Simplify
  * Symbols 
    * <
      * Less than
      * Fewer than
    * >
      * Greater than
      * More than
    * ≤
      * Less than or equal to
      * At most
      * No more than
    * ≥
      * Greater than or equal to
      * At least
      * No less than



### 5-4 Solving Compound Inequalities

  * Compound Inequalities  

    * 2 inequalities graphed on the same number line
    * 2 types 
      * "And" inequalities
      * "Or" inequalities
  * "And" Inequalities 
    * Solution/graph that is called an intersection
    * Intersection 
      * Where the 2 inequalities overlap
    * Basically connecting the 2 points with a line.
    * Solution has to be between the 2 points
    * x ≥ 2 and x ≤ 6  


    * 2 ≤ x ≤ 6  

    * {x | 2 ≤ x ≤ 6}
  * "Or" Inequalities
    * Solution/graph is called a union
    * Union
      * Typically graph lines that run opposite of one another
    * Both inequalities will always be separated by the word "or"
  * If you have the same number with arrows going in both direction it is all real numbers.



### 5-5 Inequalities Involving Absolute Value

  * 2 cases ( + case and - case)
  * For the negative case, you FLIP your inequality sign before you solve or graph
  * If value to the right of the absolute value inequality is - the answer is no solution



### 5-6 Graphing Inequalities in Two Variables

  * Boundary Line  

    * A line that divides a coordinate plane into 2 halves.
  * Closed-half plane 
    * Straight solid line
    * ≤ ; ≥
  * Open-half plane 
    * Graph a dashed boundary line
    * < ; >
  * Solution sets 
    * The ordered pair must be on the closed boundary line or in the shaded region
  * If ≥ or > shade above your boundary line.  

  * If ≤ or < shade below your boundary line.  

  * Steps 
    * Solve inequality for y (slope-intercept form)
    * Graph y-intercept
    * Use slope to get a 2nd point
    * Graph appropriate boundary line
    * Shade above or below




	6. Systems of Linear Equations and Inequalities

### 6-1 Graphing Systems of Equation

  * System 
    * 2 equations
  * Consistent 
    * Lines touch
  * Possible Solutions of a System 
    * # of solutions 
      * Terminology 
        * Graphs
    * 1 
      * Consistent
      * Independent 
        * Intersecting
    * Infinite 
      * Consistent
      * Dependent 
        * Same
    * No solutions 
      * Inconsistent 
        * Parallel



### 6-2 Substitution

  * Steps 
    * Solve 1 equation for either x or y
    * Substitute the expression in step 1 into the variable it replaces in the remaining equation
    * Substitute the value in step 2 back into the original equation
    * Give your answer as an ordered pair if possible



### 6-3 Elimination Using Addition and Subtraction

  * Steps 
    * Make sure both equations are in the same order, then set them vertically.
    * Add or subtract the equation to eliminate one of the variables
    * Substitute the value from step 2 into one of the equations and solve for the other variable
    * If possible, write your answer as an ordered pair
  * If you have coefficients with opposite signs, just combine both equations.
  * If you have the same coefficient with the same sign, change the sign to every term in 1 equation then combine.



### 6-4 Elimination Using Multiplication

  * Steps  

    * Multiply at least one equation by a consent to result in two equations that contain opposite terms.
    * Stack equations and combine like terms.
    * Substitute the value in step 2 into either equation and solve for the remaining variable.
    * If possible, write equation as ordered pair.



### 6-6 Organizing Data Using Matrices

  * Matrix  

    * A rectangular arrangement of numbers in rows and columns enclosed in brackets
  * Elements 
    * A number used in a matrix
  * Dimensions 
    * The number of rows and columns in a matrix
    * Row # x column #
  * Most matrices are named using an uppercase variable.
  * In order to add or subtract matrices, the dimensions must be the same.
  * When subtracting you must change all signs of each element in the 2nd matrix before you combine.



### 6-6 Multiplying a Matrix by a Scaler

  * Scaler  

    * Is a constant that is used to multiply each element in a matrix.



### 6-8 System of Inequalities

  * Steps 
    * Make sure each inequality is in slope-intercept form before you graph.
    * Graph each inequality on the same coordinate plane.
    * The portion of your graph where the shaded inequalities overlap is your solution area.




	7. Polynomials

### 7-1 Multiplying Monomials

  * Monomial 
    * A #, a variable, or the product of the two.
    * If your fraction has a # for a denominator, it's still a monomial. It can not have a monomial as a denominator.
    * A monomial is one term
  * Product of Powers 
    * To multiply two powers with the same base, add their exponents.
  * Power of a Power 
    * To find a power of a power, multiply the exponents.
  * Power of a Product 
    * To find the power of a product, find the power of each factor and multiply.



### 7-2 Dividing Monomials

  * Quotient of Powers  

    * To divide 2 powers with the same base; subtract the exponents.
  * Your answer goes wherever the largest exponent was at.
  * Power of a Quotient 
    * To find the power of a quotient, find the power of both the numerator and denominator.
  * Zero Exponent Property 
    * Anything raised to the power of zero is 1.
  * Negative Exponent Property 
    * If you have a negative exponent you just flip its position and change the sign.



### 7-3 Scientific Notation

  * Tips 
    * If your standard form is a whole number your exponent will be positive and decimal will move to the right.
    * If in standard form is a decimal number your exponent will be negative. Move the decimal to the left.
    * Decimals will get behind the 1st nonzero digit to the left.



### 7-4 Polynomials

  * Polynomial  

    * Is a monomial or the sum or difference of monomials
  * Types 
    * Monomial = 1 term = 3𝑥^3
    * Binomial = 2 terms = 3𝑥^3+2𝑥^2
    * Trinomial = 3 terms = 3𝑥^3+2𝑥^2+𝑥
    * 4 term polynomial
  * Degree 
    * You add each terms exponent and the largest sum names the degree.
    * Chart 
      * 0 
        * Constant
      * 1 
        * Linear
      * 2 
        * Quadratic
      * 3 
        * Cubic
      * 4 
        * Quartic
      * 5 
        * Quintic
      * 6 + 
        * 6th degree, 7th degree...
  * Standard Form of a Polynomial 
    * Ordering terms from greatest to least exponent
  * Name Polynomials 
    * Type then degree



### 7-5 Adding and Subtracting Polynomials

  * Steps to Adding  

    * Put both polynomials in standard form
    * Add missing terms into your polynomial
    * Stack and combine like terms
  * Steps to Subtracting 
    * Change signs to each term in second polynomials
    * Follow steps for adding



### 7-6 Multiplying a Polynomial by a Monomial

  * Simplify Expressions
  * Solving



### 7-7 Multiplying Polynomials

  * FOIL Method  

    * Use when multiplying a binomial x binomial
  * Distributive Property 
    * When multiplying any other groups of polynomial
  * What FOIL means   

    * **F** \- 1st x 1st
    * **O** \- outer x outer
    * **I** \- inner x inner
    * **L** \- last x last
  * Stack the two middle terms because they are like terms



### 7-8 Special Products

  * Square of a Sum
  * Square of a Difference
  * Product of a Sum and a Difference




	8. Factoring and Quadratic Equations

### 8-1 Monomials and Factoring

  * Factored Form 
    * Express the #/monomial as the product of prime #'s and variables.
  * Only begin your factored forms with 1 if the monomial is negative.
  * Greatest Common Factor (GCF) 
    * Product of the common prime factor between monomials
  * Steps 
    * Factor each monomial
    * Circle common factors
    * Multiply common factors back together



### 8-2 Using the Distributive Property to Factor

  * When to Use Distruibutive Property to Factor  

    * When you have a binomial or a trinomial.
  * Steps 
    * Find the GCF
    * Divide each term by the GCF
    * GCF goes outside the parentheses and what's left after you divide stays in ().
  * If leading coefficient is positive, GCF is positive. If leading coefficient is negative, GCF is negative.
  * Factor by Grouping 
    * Use with a four term or more polynomial
  * Steps 
    * Group together the terms that have the most in common.
    * Factor out the GCF from each group.
    * Divide each group by its GCF.
    * If () are the same, group everything outside together.
    * If () are opposite signs, factor out negative 1 before you group.



### 8-3 Quadratic Equations

### 8-4 "Still Using Magic X"

### 8-5 Quadratic Equations: Differences of Squares

  * To the 4th Power will not always mean you need to split twice



## Factoring Trinomials: the Magic "X" Method

  * Step #1  

    * Remove a GCF if possible (divide each term by the GCF)
    * Eliminate a NEGATIVE coefficient of x^2 (divide terms by -1)
  * Step #2 
    * Make sure equation is in the form ax^2+bx+c=0
    * Draw a large "X" on your paper.
  * Step #3 
    * Identify the "a" and "c" components of your given equation
    * "a" is the coefficient of the x^2 term
    * "c" is the constant term
  * Step #4 
    * Multiply (a)(c)
    * Place that value in the top region of your "X"
    * Identify "b", the coefficient of x.
    * Place "b" in the lower region of your "x"
  * Step #5 
    * Find 2 number that when multiplied = ac AND when added = b.
    * Make a list of factors of ac. Find terms that add to b.
    * Place those numbers (factors) in the left and right regions of your "X"
  * Step #6 
    * Divide both factors by the leading coefficient of x^2 which is "a"
    * Simplify the fractions.
  * Step #7 
    * Use the simplified fractions to write the trinomial expressed in fractored form. 
      * There are 2 factors!!! NOTE: for each fraction... 
        * The BOTTOM number (denominator) is the coefficient of "x"
        * The TOP number (numerator) is the constant.
      * Complete: ( )( )
  * Step #8 
    * CHECK your answer by multiplying the terms using the "foil" method.




	9. Quadratic and Exponential Functions

### 9-1 Graphing Quadratic Functions

  * Quadratic Function 
    * Parent Function: 
      * f(x) = x^2
      * y = x^2
    * Standard Form: 
      * f(x) = ax^2 + bx + c
    * Type of Graph: 
      * Parabola
    * Axis of Symmetry (AOS): 
      * x = -b/2a
    * Y-intercept: 
      * c
  * Vertex 
    * Highest or lowest center point of a parabola
  * AOS 
    * A line of symmetry that passes through the vertex of a parabola splitting the parabola into 2 mirror images
    * Always the x-coordinate of the vertex
  * a < 0 
    * Minimum
    * Lowest point
  * a > 0 
    * Maximum
    * Highest point
  * Graphing Quadratics 
    * Steps 
      * Find the AOS.
      * Find the vertex by plugging in the value of x in step 1 into the equation and solve for y.
      * Find the y-intercept (value of c in standard form)
      * Mirror the y-int across the AOS.
      * Connect the points with a smooth curve




End file.
